A conventional 3rd generation Computed Tomography (CT) system with a single circular source trajectory is limited in terms of longitudinal scan coverage since extending the scan coverage beyond 40 mm results in significant cone-beam artifacts. A multiaxial CT acquisition is achieved by combining multiple sequential 3rd generation axial scans or by performing a single axial multisource CT scan with multiple longitudinally offset sources. Data from multiple axial scans or multiple sources provide complementary information. For full-scan acquisitions, we present a window-based 3D analytic cone-beam reconstruction algorithm by tessellating data from neighboring axial datasets. We also show that multi-axial CT acquisition can extend the axial scan coverage while minimizing cone-beam artifacts. For half-scan acquisitions, one cannot take advantage of conjugate rays. We propose a cone-angle dependent weighting approach to combine multi-axial half-scan data. We compute the relative contribution from each axial dataset to each voxel based on the X-ray beam collimation, the respective cone-angles, and the spacing between the axial scans. We present numerical experiments to demonstrate that the proposed techniques successfully reduce cone-beam artifacts at very large volumetric coverage.

Since 1990, multislice or multidetector-row computed tomography (CT) systems have become the standard CT architecture for premium medical scanners: the detector has multiple rows, that is, a 2-dimensional array of detector cells, yielding a cone-beam geometry. Since CT systems with this geometry do not generate the rays to be perpendicular to the rotational axis, 2D image reconstruction algorithms result in reconstructions that suffer from cone-beam artifacts. For axial scan mode where the table does not move during gantry rotation, Feldkamp, Davis, and Kress proposed a 3D cone-beam reconstruction algorithm (FDK) that is an adaptation of the 2D fan-beam filtered backprojection (FBP) to a cone-beam geometry [

In helical cone-beam scans, the data are fundamentally complete, provided that the helical pitch is not too high. Therefore, exact reconstruction can be achieved [

Other interesting reconstruction problems have been posed by the demand for dynamic object imaging, such as cardiac CT. Theoretically, a 2D object can be reconstructed accurately if all its line integrals are measured at least once. This condition leads to the notion of half-scan or short-scan, which means that the scan interval per acquisition is only

In axial scan mode, conventional 3rd generation CT systems suffer from increased cone-beam artifacts with increasing coverage, due to incomplete data and suboptimal processing of the available data. The efforts to increase axial scan coverage without sacrificing image quality led to the weight-based cone-beam reconstruction algorithms for single circular trajectory acquisitions [

In Section

In this section, we briefly discuss three possible multi-axial CT acquisition system architecture concepts and the corresponding full-scan and half-scan acquisition and reconstruction modes.

First, the simplest way to acquire multi-axial data without modifying any system hardware is to take a series of axial scans sequentially (Figure

Multi-axial acquisition system concepts: (a) multiple sequential axial scans taken by single source 3rd generation system, (b) a system with multiple sources distributed longitudinally, and (c) a multisource inverse geometry source (MS-IGCT) with 2 by 10 area source. Each subsinogram can be rebinned to conventional 3rd generation sinogram.

Second, a multi-source system with longitudinally distributed sources or a line-source CT is another way to acquire multiple axial data. Two or more focal spots are distributed along the z-axis and are alternatively emitting X-ray. There is no spatiotemporal misregistration between the two (or more) axial datasets, which is preferable for dynamic applications (Figure

Third, a multi-source inverse geometry CT (MS-IGCT) consists of a number of focal spots distributed in x (trans-axially), each emitting a relatively narrow X-ray beam through a small portion of the field-of-view, shown in Figure

In this paper, we present analytic cone-beam reconstruction algorithms for multi-axial acquisitions. While we focus on the line source CT case, the methods and results translate directly to any other type of multi-axial acquisition.

The multi-axial acquisition architectures described in the previous section has two scan and reconstruction modes to meet various clinical needs:

Multi-axial acquisition geometries with 3 longitudinal acquisitions (source positions marked as red dots): (a) fully opened collimation, (b) full-scan mode with semi-closed collimation (central source is shown in opposite side.), and (c) half-scan mode with fully opened collimation.

The most straightforward way to combine data from multi-axial acquisitions is to reconstruct the volumes corresponding to each single axial acquisition separately and computing a weighted combination of the reconstructed slices from the different axial datasets. In practice, most of slices will be substituted with the slices reconstructed from the single axial dataset at same

In half-scan mode, conjugate rays are in general not available. However, just like for full-scan, the desired volume can be reconstructed by taking a weighted combination of the reconstructions from each axial dataset. Again, a sufficiently large detector illumination is required to avoid any gap in the volumetric coverage, as shown in Figure

The multi-axial CT architecture introduces additional X-ray sources or scans, and the concerns on additional X-ray dosage should be addressed. Instead of discussing absolute X-ray dose which can be traded off with the image quality, we would like to focus on the dose-efficiency to utilize the best out of the given X-ray dosage. To achieve the maximum dose efficiency, first every X-ray from the tube should be detected. The multi-axial CT architectures described in previous section with big enough detector will certainly meet this criterion.

Second, every detected projection should be used in the image reconstruction process. In other words, we can achieve the maximum dose efficiency by only acquiring the projection data needed in the reconstruction process. The detector utilization of the multi-axial CT acquisitions and reconstruction algorithms are shown in Figure

The detector utilization of the FDK-based and TOM windowing reconstruction approaches. The area closed by red lines represents the actual acquired area of the detector for (a) multi-axial scan/line source CT, (b) line source CT curved collimator, and (c) multi-source inverse geometry CT. The area in blue represents the data required to reconstruct volume using TOM windowing approach. Note that TOM windowing has an advantage over FDK in detector utilization if data is acquired by the line-source CT with the collimator or the multi-source inverse geometry CT.

Finally, a uniform flux profile should be achieved along the reconstruction volume to maximize dose efficiency since the nonuniform flux/noise profile degrades the dose-efficiency. The flux uniformity of the various CT architectures and associated reconstruction approaches are shown in Figure

The flux uniformity of the various CT architectures and associated reconstruction approaches: (a) 3rd generation CT with FDK reconstruction, (b) line source CT with FDK slice substitution approach, (c) line source CT with TOM windowing reconstruction, and (d) multi-source inverse geometry CT with TOM windowing approach. The areas with different flux level are shown in different colors.

In helical reconstruction algorithms, a so-called Tam-window is used, named after one of the inventors, Kwok Tam [

One way to use the tessellation property is to synthesize plane integrals. First the Radon transform of the projections is computed, and the derivatives are taken to apply Grangeat’s theorem. Tam’s tessellation approach is then used to patch the triangular regions. Addition of a boundary term is required for more accurate (but still nonexact) reconstruction. Finally the desired volume can be reconstructed by taking a 3D inverse Radon transform. Note that not all plane integrals can be computed due to the nature of the multi-axial architecture, but missing integrals can be estimated by interpolation. A second way to apply the tessellation approach is using filtered backprojection, which is our preferred approach. We apply an existing filtered backprojection technique, such as FDK, and we combine it with a TOM-windowed backprojection. Geometrically, the cones from complimentary sources can match perfectly with each other so that data between the TOM window boundaries jointly cover the entire volume. This means that rays from adjacent sources should intersect each other at the isocenter, as shown in Figure

Efficient TOM windowing: during the backprojection of each voxel and each view, it is decided if the projection of that voxel falls inside the TOM window and if not, the contribution from that view is discarded. 3 longitudinal sources are marked as

The windowing can be achieved by applying a binary mask to the filtered projection data, but this approach suffers from quantization artifacts. A better way to implement the TOM window is to make a decision during the backprojection of each voxel for each view, by determining whether or not the voxel analytically projects inside the TOM window. This implementation has less quantization error but higher computational complexity. Figure

View contributions from source

The detector region surrounded by the TOM boundaries for a given source is selected by a weight function

As described above, TOM windowing is implemented during the backprojection step for each longitudinally located source. The backprojected volume associated with a given source will be partial, and the reconstruction step will be completed with the summation of all partial volumes. Thus, the final reconstruction is given by

The FDK algorithm employs a ramp filter in its filtering step. This works well for a full axial scan because the boundary terms incurred by the difference between a fan-beam geometry and a parallel-beam geometry are canceled out by backprojecting over a full rotation [

The view derivative can be computed at interlaced sampling locations, as described for a helical trajectory in [

Since only a finite number of discrete view samples will be taken during real scanning, the binary windowing operation can introduce artifacts, which are also discussed in [

Discrete view sampling: true pi-line, (a) in solid line and approximated pi-line, (b) in dashed line, due to the discrete view sampling. Note that pi-segment associated with (b) is now smaller.

This problem can be mitigated by an additional linear smoothing step. Instead of taking binary weights near the TOM boundaries, partial weights are determined by how close projected voxels are to the boundary. One could define a fixed width region along the TOM boundary, and whenever voxels are projected into that region, they will receive a partial back-projection contribution instead of a full contribution. However this approach will not result in desired smoothing effect because the weight at a certain source position and the weight at the corresponding conjugate source position might not add up to one. Instead, we define a fixed longitudinal interval around each voxel and project (the endpoints of) this interval onto the detector. This way the conjugate weights contributing to a given voxel will add up to one.

The TOM windowing approach described above combines contributions from adjacent source positions. However, it can be extended to combine contributions from source positions that are not immediately adjacent.

In half-scan mode, triangular patching of conjugate data from longitudinally offset axial scans is no longer possible. Combining data from multiple longitudinally offset axial scans becomes challenging because the reconstruction volume is now divided into several regions: regions with no illumination and regions illuminated once, twice, or three times, which means that complementary information is not always available. Various weighting approaches utilizing complementary information have been proposed by introducing the scale factors representing how much information each projection contributes to a given voxel [

We propose a new approach where the weights are computed based on the cone-angle on a voxel and per view basis. Figure

Multi-source half-scan regions at a given fan angle: (a) a region illuminated by only (

The actual implementation of the cone-angle dependent weighting approach is a bit more complicated, first because some voxels will not be illuminated from any source at all, as in region (d) in Figure

To minimize discontinuities along different regions of reconstructed volume and still achieve cone-angle dependent weighting to effectively combine multi-axial acquisition data, feathering is required for smooth transition. The weight

Each region in Figure

To make a smooth transition from region (a) to region (c) and from region (b) to region (c), we propose two approaches: (i) an approach with fixed feathering width and (ii) an approach with fixed slope but variable feathering width. We present a specific example in Figure

The top picture represents a longitudinal view of the proposed weighting scheme. Two specific slices are selected: slice

Region (d) is treated separately because there is no ray passing through from any source. In this region, we extrapolate the detector data and make the weight

With a limited number of longitudinal samples, this approach can still result in transaxial discontinuities. A simple way to avoid this is to oversample in

To evaluate the proposed algorithms for multi-axial acquisitions, two conventional 3rd generation CT geometries and two multi-axial acquisition geometries are investigated, as presented in Figure

Illustration of proposed CT geometries: (a) and (b) are 3rd generation CT systems with 40 mm and 80 mm coverage, and (c) and (d) are multi-axial acquisition CT systems with 120 mm and 160 mm coverage. Note that the coverage of a multi-axial acquisition system is the same as detector size. 3rd generation system requires bigger detector for the same scan coverage.

Note that for 3rd generation CT system the scan coverage is defined as the projection of the physical detector to isocenter. On the other hand, the scan coverage for the studied multi-axial acquisition system is equal to the size of detector, which means that more coverage is obtained with the same size of detector, as shown in Figure

Figures

Worst-case images from (a) 3rd generation CT system with 40 mm coverage, (b) 3rd generation CT system with 80 mm coverage, reconstructed with FDK, (c) multi-axial acquisition CT system with 120 mm coverage, (d) multi-axial acquisition CT system with 160 mm coverage, reconstructed with slice substitution approach, using FDK slices, (e) multi-axial acquisition CT system with 120 mm coverage, and (f) multi-axial acquisition CT system with 160 mm coverage, reconstructed with TOM window-based reconstruction. Grayscale: (−50 HU, 50 HU). Note that the worst-case slices for 3rd generation system are the slices located at the edge of scan FOV, and the worst-case slices for multi-axial acquisition system are the slices located in between two sources.

All images are displayed in tight window, (−50 HU, 50 HU). The results show that the cone-beam artifacts are very severe in the 3rd generation CT geometry. Images from FDK-based slice substitution approach show some residual cone-beam artifact (Figures

Figures

Figures

Worst-case images from (a) 3rd generation CT system with 40 mm coverage, (b) 3rd generation CT system with 80 mm coverage, reconstructed using FDK with Parker weighting approach, (c) multi-axial acquisition CT system with 120 mm coverage, (d) multi-axial acquisition CT system with 160 mm coverage, reconstructed with slice substitution approach using FDK parker weighting slices, (e) multi-axial acquisition CT system with 120 mm coverage, and (f) multi-axial acquisition CT system with 160 mm coverage, reconstructed with cone-angle dependent weighting reconstruction. Grayscale: (−50 HU, 50 HU). Note that the worst-case slices for 3rd generation system are the slices located at the edge of scan FOV, and the worst-case slices for multi-axial acquisition system are the slices located in between two sources.

In half-scan mode, the cone-beam data is always much less complete than in full-scan mode, specifically because conjugate rays are not available, and hence image artifacts are also much more severe. Figures

Since TOM windowing approach is based on the FDK type of implementation, additional correction approach such as radon space-based correction [

In this paper, we present multi-axial CT acquisition geometries, which can be implemented by performing multiple axial scans with a single source 3rd generation system or by performing one or more axial scans with a multi-source CT system, in which sources are offset longitudinally. We propose corresponding reconstruction algorithms for full-scan and half-scan protocols. Both the TOM windowing reconstruction algorithm for full-scan mode and the cone-angle dependent weighting reconstruction algorithm for half-scan mode successfully reduce cone-beam artifact compared to a 3rd-generation CT acquisition with a single circular trajectory, even when the scan coverage is increased up to 160 mm. The same techniques can be applied to an inverse-geometry CT system.

Multi-axial CT geometries offer additional benefits associated with the reduced cone-angle, such as reduced Heel effect and reduced scatter, and there is room to optimize the target angles for each of the longitudinally offset spots. We have published some work in this area in [

This work was supported in part by the National Institute of the Health under Grant 1R01EB006837.